Their last campaign was their best yet as they beat Munster 31-13 back in May to become the first Scottish team to win the GUINNESS PRO12.
Matters in Europe however have not proved as fruitful, having failed to make it through to the knockout stages for the last nine seasons.
They kick off their new campaign on Saturday with a trip to French outfit Racing 92 in Pool 3 and Seymour admitted they were setting the bar high for themselves this time around.
"Our ambition is very high. We weren't private about our ambitions in the league last year," Seymour told the Glasgow Evening Times.
"Gregor Townsend was very vocal about the way he viewed the league. External pressure is something we don't worry about it but as a squad it [reaching a European quarter-final] is something we badly want to do, especially after coming so close last year.
"The guys have played in Europe enough to know what to expect. There is confidence amongst this group of players that this is something we need to do, to step up to.
"We know if we play the rugby we can really play we can turn over the group and progress."
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