Testosterone gel goes on the skin, and skin touches other people. That makes this a sensible beginner question, and the answer is short.
Yes, while the gel is wet
Yes, skin contact can pass it on. my.clevelandclinic.org states: It’s important that you don’t transfer the gel to another person through skin-to-skin contact while the gel is wet.
Free background reading
For background on hormone therapy in men, see Healthline on hormone replacement therapy for men and a second piece on HRT for males, its benefits and risks. A narrative review of testosterone replacement therapy in men aged 50 and above covers safety considerations in more depth.
The cheap step and the step to leave to a clinician
The cheap step is to write down who shares your home and bring that to your appointment. How and when to apply the gel, and what to do after, is the clinician's instruction to give, not something to piece together from a search.
Ask in your own words
If a licensed clinician prescribes gel, ask exactly how to avoid passing it on. A good one will answer without making you feel silly.