Warriors may sit rock-bottom of Pool Four with four defeats from four outings, but former Ulster winger Seymour is keen to preserve a growing momentum that has developed from an 1872 Cup double over Edinburgh and a hammering of Treviso.
And, with Gregor Townsend's team having already lost twice to the Ulster men this term in both PRO12 and European action, the Warriors stalwart is determined to gain revenge over Mark Anscombe's men as Glasgow look to salvage some pride.
Seymour said: "We have come back strongly from a disappointing run and, after winning home and away against Edinburgh and putting together a record win over Treviso, it is vital we kick on at Ulster. It doesn't matter what the tournament is, we want to win and we owe Ulster for a couple of defeats. We will be throwing everything at them.
"The results have not been what we would have liked in the Heineken Cup, but we have two games left at Ulster and then back at Scotstoun against Northampton. It is vital we keep our momentum going so that we turn back around into the PRO12 with our confidence high.
"We are under no illusion as to how tough it will be against Ulster but there is a real hunger and determination to go over there and get a result."
The 24-year-old has sizzled with six tries from eight outings this term and touched down in Warriors' 41-7 record victory over Treviso last time.
But Seymour has warned there can be no drop in standards against the Ulster men.
He said: "We are now sitting second in the PRO12 table. To a certain extent it is frustrating that we are now switching to the Heineken, but it is all about winning in every game you play in.
"We have always found it tough against Treviso – they beat us at Firhill last year. So it was a massive performance last time and, with the bonus point, it has taken us into second place on our own. But there are still areas we can improve upon.
"You look at how Ulster put almost 50 points on the board against the Scarlets. But we went to Ravenhill on the opening day of the season and only lost on a breakaway try and that is one that still hurts."
Meanwhile, former Scotland captain Mike Blair has retired from international rugby at the age of 31. The nation's most-capped scrum-half represented his country on 85 occasions, 14 of them as skipper.