Are there other character-specific sidequests except the ones of Garrus and Wrex?

The only other character who has a quest like this in the original Mass Effect is Tali, but it's less involved than either of those. You'll find some data regarding the Geth while finishing another (optional) mission on an uncharted planet.

You have the choice of either giving it to Tali, ensuring that she has something to bring back from her pilgrimage, or you can just be a jerk and keep it. This has no real consequences either way, even in future games.

More details at the Mass Effect wiki.

Garrus

Has one where you go after an Organ black market dealer.

Wrex

You go after a family battle armour

Tali

You find Geth stuff

Liara

I don't know

Ashley

Just have her with you when you talk to the guy who wants his wife's body back

Kaidan

I dont know

Tali sort of has a personal quest. During UNC: Geth Incursions, Shepard can find a cache of data in a terminal detailing the early history of the Geth. If you give this to Tali, she will use it as a gift from her Pilgrimage.

See the Mass Effect Wiki for more information.