I am nominated to Community Contribution Award. Should university financially support my travel? I have been nominated for a community contribution award. I was thrilled to be considered for this award. The conference also has workshops hardcoded to my research.

However, while discussing with my supervisor, he seemed hesitant that my travel could be financially supported, and he said if it a publication, it would be better.

He asked to send him email, it seems disappointing his reaction seemed disappointing. He said you cannot travel to all-conference, although I did not travel to any conferences since my PhD start?

My question: Should program financially support my travel to present at workshop plus being award nominee?
Sadly, what the university should do, what it can do, and what it is willing to do might all be different. 

In a perfect world, yes, they should fund you. However, there may be limited funds or even regulations that bind their decision. Also, your relationship with the supervisor might be an issue if they don't think that this is worthwhile for you or them. Or even, if it would be your supervisor providing the funding under a grant, the grant itself might have rules. Finally, note that not every university will provide funds for students in any case. It might be harder to justify an unusual expense than a common one. 

It wasn't exactly clear if you sent the email you were already asked to send and were then refused or if you haven't done that yet. Don't push the issue to the point that your advisor is angry and unyielding, but a formal request is probably appropriate. But try to make your case on the merits of the award, both for yourself and the university. And, there may be things of direct interest to your current studies at the conference also. 

And if you have to self-fund for this, it may still be worth it for you in the long term if you use the time well at the conference to extend your circle of contacts for the future.