Do Armenians and Georgians belong to the same language family?

No. Armenian and Georgian languages are quite different.

Armenian language belongs to Indo-European languages' family, consisting a separate branch thereof. While Georgian language is Caucasian language and has been for centuries as only Caucasian language, having alphabet and literature.

Georgian and Armenian languages are written by their own scripts. Georgian alphabet was invented by King Parnavaz of Iberia in V c BC. The oldest Georgian inscription found ever belongs to II AD. Georgian alphabet has changed its forms for 2 times and nowadays Georgian language has 3 different alphabets. First 2 is used for church services and the newest is the official script for Georgian language as a state language of Georgia.

Armenian alphabet was invented by Armenian scholar Mesrop Mashtots in V c AD, 10 centuries later. It has never changed forms, only some letters were added in later mid centuries and it is used as official script for Armenian language.

Davit Chikvaidze