「グジュグジュ」 is indeed an onomatopoeia describing something being wet, watery, damp, etc. (often, if not always, in a grubby way.)
Onomatopoeias often have numerous variants: therefore, many of them cannot be found in dictionaries. We also say 「グチュグチュ」、「グショグショ」、「グチョグチョ」, etc.
Here, it is talking about the inside of a dog's ears by using the onomatopoeia. This particular dog has wet discharge inside its ears, which is not ideal as that can help bacteria thrive in them.
So, the first two lines mean:
"(As for) ムー the cocker spaniel.."
"It is wet and grubby inside his ears. Gotta do something to get rid of that, don't we? "