I cannot answer the question of whether the field of $q$-series is dead. I would make this a comment, but I lack sufficient reputation.

I am a Banach space theorist, and Banach space theory is not fully dead. However, it is sufficiently unfashionable as to make it terribly difficult for me to find a job (even with a very strong research record). In fact, I have been completely unable to find employment, and I am now forced to leave academia because of it. Granted, geographical factors, personal connections, and luck may all be different for you. However, I made the mistake of choosing my research area only because of its intellectual interest to me, without any practical consideration of the job market, and I would caution you against doing the same. If your advisor says the area is dead, I would listen.

I would also say that, if you like that area, you can study it as much as you want after you find a permanent position.