Valencia, who started last season's group stages with a thumping 4-0 away win over Bursaspor, came the closest to netting a winner when Adil Rami's overhead kick struck the crossbar midway through the second half.
The result leaves the two teams in joint second place in the standings, two points behind Chelsea who were 2-0 home winners over Bayer Leverkusen.
Valencia started their domestic campaign with back-to-back victories and they looked promising early on Tuesday night with Rami heading over before Roberto Soldado, who has struck four times in La Liga already this season, also missed the target with a header.
Summer signing Pablo Piatti then tried his luck with an effort that was straight at home goalkeeper Laszlo Koteles.
Mehmet Topal, Dani Parejo and Piatti also had efforts without success before Belgian champions Genk started to fight their way back into the game with Daniel Pudil volleying wide and Jeroen Simaeys bringing a save out of visiting goalkeeper Diego Alves.
Jelle Vossen also drew a parry out of Alves just before the teams went off for half-time.
The second half saw Valencia again start brightly with Piatti shooting over before Soldado, who had netted 16 times in Los Che's previous 11 matches, saw a cross-shot parried by Koteles in the 55th minute.
The match looked to be heading towards a goalless stalemate but Valencia came within inches of breaking the deadlock in the 72nd minute when Pablo Hernandez's cross was met by a spectacular overhead kick from Rami, but the France international was unlucky to see his effort strike the woodwork.
Valencia would not come any closer than that in the final stages of the match, but Genk might have snatched the points for themselves in the 83rd minute.
Nadson was almost the Genk hero but his shot was tipped over by Alves as the match finished level.