If you have very fast internet connection (faster then 100mbps) then yes a new motherboard will help with internet speed.
What is probably the more underlying factor here is that you have an old ddr2 motherboard that is 7-10 years old that has 7-10 year old hardware in it. Even worse if regular maintenance has not been done on that computer then it has a ton of junk files, registry keys, fragmented hard drive, and tons of startup programs that are slowing it down which is what is making everything so slow.