Can telogen effluvium which happens due to crash diets cause androgenic alopecia?

Yes, for some people a shedding episode (telogen effluvium) can precipitate androgenetic alopecia. It is not correct to say it caused it. No. The individual was likely to develop AGA at a later date in life- the shedding episode simply sped it up.

a crash diet of more than twenty pounds without the proper nutrition or supplements can expedite hair loss. If it is general shedding the lost hair may come back. However if you have family history of hair loss and the hair loss pattern that you get is androgenic alopecia the chances of hair growing back is very low. 

Yes, crash diets can cause telogen effluvium, more so in women than in men.  Women who develop TE are often prone to this condition.