Can weight lifting speed up hair loss?

In general regular exercise isn't something that would cause hair loss. However, some supplements may affect your hair and steroids definitely will. Since you aren't taking any of those I wouldn't expect weight lifting to affect your hair density/thickness. 

It’s understandable why this myth is perpetuated as testosterone is very much part of the hair loss story and some weight lifters may be alpha males with higher levels of testosterone - especially if they take supplements which, of course, is not recommended. Considering that androgenic alopecia (male hair loss pattern) by age 50 affects over 50% of the male population, it follows that 50% of weight-lifters will also be affected. Weight-lifting will not accelerate familial hair loss - which is treatable especially if therapy is begun early on.   

Weight lifting will not accelerate your hair loss unless you take steroids along with your vitamin supplements