Yes it's a Fishtail Palm. Caustic seeds on these guys so be wary of handling them as they begin to develop later on. Also, the seeds start being produced from the base of this palm first then work their way in reverse up the palm eventually reaching the top and ending that trunks life! :}
What does the seeds look like? I don 't know how old this plant is. I just brought it home from work. Can you cut off the branches?
You don't need to worry for a little while as the plant needs to reach a mature age before starting to flower. Your plant if still quite a few years away from this. The seed will turn a bright red colour as it ripens and it's probably best to just cut of the flower spikes (spadix) before seed sets. Because it looks like you have Caryota mitis it is a clumping palm so any trunks you remove will be replaced by new ones eventually re-shooting from the base. I'd recommend a reasonably sheltered spot with plenty of good fertilser high in nitrogen in the warmer growing months. Good drainage and plenty of water again when growing. :}
