Yes,
someone coming on a Family Visit Visa can apply for a Driving License.

Validity of the license = until expiry of the Visa. (Hence upon passing the test, extend the visa until its maximum duration and then pay for the Driving License Card)

If they hold their home country license, they will be permitted for a Half Course of 20 days for QAR 2800.. Full course is for 40 days and for QAR 3300