No, your "full coverages" of collision and comprehensive would not cover the expenses for a transmission replacement unless your vehicle was in an accident covered by your insurance policy which caused the damage to your transmission. Mechanical problems not caused by damage from an insurable incident are not normally covered by an insurance policy.
Typically coverage for a transmission, and other mechanical issues not related to an accident or other insured incident, is covered under warranties or third-party additions.
Coverage for specific parts of your vehicle, such as the engine or transmission, to be repaired or replaced is usually done with by means of a warranty. Warranties can be purchased at dealerships, financial institutes or online. Some financial entities also offer mechanical breakdown insurance.
If you have either a warranty covering your transmission or mechanical breakdown insurance then you can see how to go about having your transmission replacement paid for by contacting the company which you have this coverage through.
If your transmission needs repaired or replaced due to damages sustained in an accident then you can contact your insurance provider to place a claim through your physical damage coverages.