The following classes would be considered as mages in Dragon Age. Additionally all classes listed under "Mage" do not have proficiency in armor or weapons besides quarterstaffs unless a subclass gives you access to it (or if you discuss a reason with your DM first). Also mages in Dragon Age usually cast bolts of magic from their staffs, you can flavor this as casting cantrips from your quarterstaff.
The term "cleric" has significance in Thedas due to its relationship with The Chantry, but "clerics" in Thedas are not divine spellcasters. One of the major differences between divine and arcane spellcasters in D&D is that divine casters have access to healing spells, whereas arcane spellcasters generally do not. Divine magic does not really exist in Dragon Age and as such most magic simply comes from The Fade. Depending on the domain of the cleric, their abilities in Dragon Age would most likely be explained as them taking on the role of healers or warrior mages. They could also be depicted as warriors who have bonded with a spirit of The Fade to aid them and their allies in battle.
Though there are no "druids" in Thedas, the abilities of the Druid class can be found in the Dalish keepers and the Shapeshifter specialisation. The Circle of Magi denies the ability to change one's shape into that of an animal exists, but Morrigan in Dragon Age: Origins proves this to be untrue. If a player chooses to play a druid it is most likely that they are an apostate who has been trained away from the Circle, or are one of the Dalish who learned the ability from someone in their clan. However it is entirely possible that a Circle Mage discovered the ability to shapeshift when studying ancient tomes in one of the mage colleges.
The term "sorcerer", "wizard", and "mage" are interchangeable in Thedas. As such a character playing a Sorcerer can simply be considered a mage with a unqiue way of casting spells. However, the "Sorcerer" class from D&D and its versatility compared to the Wizard class makes it a candidate to be depicted as a Blood Mage. If a player chooses to play a Sorcerer then you may choose to reflavour the Sorcery Points that the class gains at level 2 as their "Blood Pool". Mechanically the limitations on the number of points a character has and can use can be explained away as the character only being able to use a certain amount of blood and still remain standing.
There are no "warlocks" per se in Dragon Age. However the theme of forging a pact with another being in order to gain power is seen a number of times in the Dragon Age games. Options and lore explanations for the Warlock pacts from the Player's Handbook are presented below. If you are using pacts from other sources these should give you an idea for how to flavour them. Warlock's are seen as having a protective force or unusual aura surrounding them while in The Fade, which can be seen by other spirits and demons. Not wishing to anger this powerful patron, these spirits do not bother the Warlock most of the time.
Archfey. The lost lore of the Elves, or Elvhenan, and the mysteries of Arlathan provide the perfect fuel for a character who chooses the Archfey pact. Powerful elven spirits from The Fade or perhaps ancient elves once sleeping in the state of Uthenera (the "long rest") might offer power and secrets from the past. Perhaps even one of the elven gods, such as Fen'Harel or Mythal, gifted the character with power to fuel the rise of the elven people.
The Fiend. The pact of the fiend is one of the easiest to transfer to the Dragon Age setting. This pact can represent a pact with a malevolent demon that for some reason has allowed the character a piece of its power without attemptiing to take over its body. The pact may also come from The Fade or may even come from the character binding themselves to the darkspawn or Archdemon, thereby gaining access to Blight Magic.
The Great Old One. In Thedas a pact with The Great Old One may represent a character seeking out one of the Old Gods (the beings that are corrupted by the Blight to become Archdemons) and trying to tap into it's power. Alternatively, the unknowable nature of this pact opens up the opportunity for a character to be bound to a previously unknown creature of The Fade, an ancienct and lost god, or perhaps they have tapped into the mysterious substance known as Red Lyrium.
With the exception of not having access to healing spells, the Wizard is the most direct representation of Dragon Age's "Mage" class. The nature of the Wizard and their reliance on study means that it is the best fit for a traditional Circle Mage. If you wish you can rule that any character playing a Wizard is, or has been, a member of the Circle.
Though there is a "Bard" specialisation in Dragon Age, their abilities are limited to inspiring allies and weakening enemies and they are not spellcasters. Though the Bard class in D&D has some similarities to the Dragon Age Bard, its spellcasting ability sets it apart. The ability to use these spells can be explained in a few ways. Rather than considering the bard a "Mage", an option is that the character has bonded with a spirit of Joy or Creativity and their abilities stem from that union. Another option would be to reflavour the spells to appear as more mundane talents of inspiring allies and demoralizing enemies.
The Ranger in D&D is not too dissimilar to the Ranger specialisation presented in Dragon Age. Both are hunters and trackers who are at home in the wilderness and have an affinity with wild animals. The major difference between the two is that the D&D Ranger also has access to spells, though they are a "half-caster" class. The ability to use these spells can be explained in a few ways. The most obvious option is for the spellcasting ability to be fueled by a spirit of the wild or one of the elven gods, possibly Andruil, Goddess of the Hunt. Another option would be to reflavour the spells to appear as more mundane talents or heightened senses that the Ranger has developed through years of training.
The Rogue class in both D&D and Dragon Age are almost identical in what they represent and how they are intended to be played, with both even having a Backstab/Sneak Attack ability. The only aspect that needs to be explained is if a player chooses to play an Arcane Trickster. Like the Ranger mentioned above, these abilities can be passed off as stemming from one of the elven deities, a spirit of The Fade, or perhaps even the maniuplation and use of lyrium.
The Barbarian class exists in Dragon Age in the form of the Berserker specialization. This style of fighting is most popular amongst the dwarves but can also be found amongst the Avvar of the Frostback Mountains on the border of Ferelden and Orlais.
Like the Rogue's relationship to the Rogue from Dragon Age, the Warrior of Dragon Age can be directly represented by the Fighter class in D&D. Like the Arcane Trickster too, the Eldritch Knight is an archetype within the class that wields magic. Eldritch Knight is quite similar to the Spirit Warrior specialization found in Dragon Age and as such a warrior having these types of abilities already has a place in the Dragon Age lore.
Nothing quite like the Monk class has appeared in the Dragon Age setting as yet, but that does not mean they should not be used. A monks abilities can be explained as a warrior mastering the art of hand to hand combat to a point that it seems supernatural, or it may actually be supernatural as the monk is fueled by a spirit (benevolent or otherwise) of The Fade.
The most direct comparison to the Paladin class that can be found in Thedas are the mage-hunting Templars of the Chantry and the knightly Chevaliers of Orlais (though the Chevaliers do not possess magical ability). Templars in Dragon Age are considered protectors or jailors depending on your point of view. Through the maniuplation of lyrium they are able to defend themselves from magical attacks and are tasked with hunting down mages. If a player wishes to play a Templar then the Paladin class (or alternatively the Elritch Knight arhcetype of the Fighter) could be suitable. Whichever class the player chooses, the Mage Slayer feat is highly recommended.