So your plugging 2 sdi feeds into their studio mixer ?

If so they would need to provide you with a reference tri-sync line that you would plug into the sync input of your Quad card, usually the bottom connection. It then synchronises with their system, not the other way around.

If you look at the decklink setup application it will show you when a reference sync is detected. Of course you will need to be running vMix at the same frame rate and resolution as their studio. Ask to talk to their technical director and tell him what you have.

Vmix and decklink will add about 6 frames of delay to the sdi outputs so if there's audio with these feeds they should use sdi embedded audio that will be equally delayed or the studio audio operator would need to delay the audio.

We have our Vmix synced to Pal black and it appears as a synchronised source on our studio mixer.
It's not hard to do but they provide you with the tri-level sync feed. Your decklink card will also accept a blackburst reference but tri-level is more stable.