Does Celeborn know that Thranduil is friends with the Avari?
Is Celeborn 100% certain that Thranduil is friends with the Avari? No, he isn't. Nobody outside of Greenwood and the Avari Tribes they speak to are 100% certain of their relationship.
However, while Celeborn might not be renowned across the lands for excessive wisdom like his wife is, he’s also not an idiot. He knows that on occasion both Oropher and Thranduil have received support of some kind from outside their kingdom, (usually in the form of Supplies, Advice, or Shared Knowledge) and never from anybody that Celeborn is aware of. He also knows that pretty much the only time one of Thranduil’s elves will stray any distance from the borders of the forest is when they're going to visit somebody, and if no Wood Elves turn up at Imladris, Lothlorien, or Lindon…… that doesn’t really leave many other options other than an Avari tribe for them to have gone too.
Celeborn doesn't get any form of confirmation on whether he’s correct that Greenwood has a friendly relationship with at least one of the Avari tribes until he shows up at Dol Guldur at the end of the War of the Ring and discovers that several tribes have already been fighting alongside the Greenwood Elves for several weeks.