Air India (AI 984) flies direct to Goa via Mumbai, with 1 hour stop at Mumbai (MUM). It's a 5 hours flight from Dubai to Goa. Probably that's the next fastest way to travel from Dubai to Goa.
All other alternatives include a change of flight. Check out one of the Dubai to Mumbai flights. There are quite a few daily flights from Dubai to Mumbai. So is the domestic connection frequency between Goa and Mumbai.
Also keep in mind the hassles of changing from the international terminal to the domestic terminal at Mumbai, from where the Goa flights operate. It's a 1 hour+ affair, at least includes a bus transfer.
Another smart alternative is to fly from Sharjah (SHJ) to Goa. There is an Air Arabia flight (G9 492) that flies direct from Sharjah (SHJ) to Goa (GOI). It's a 3 hour flight to Goa.
Mumbai is well connected with Goa. If you could to find a direct flight to Goa, fly to Mumbai and get a connection to Goa.