Thanks for your question. First things first, let me tell you that I'm now posting as myself, Chris Anderson, rather than as tartugas.
As for your Q, it would take approximately 20 minutes to get to JJ from Long Island City by cab. But the challenge can be getting a cab over there. Cabs are not really as plentiful in the outer boroughs.
If you are staying anywhere near the N/R/Q trains, however, it's a relatively quick ride to 57th St and 6th avenue. From there you can walk over to JJ in about 10-15 minutes, or hop on the crosstown M57 bus which runs often and would get you there in a little less time. JJC is at 59th and 10th Avenue