I feel like there are no concepts we cannot define, theoretically at least. Is there any argument suggesting the exact opposite? How strong is that argument?
Yes and No.
Every concept can be defined. Lets take for example concept "square-circle". Sounds stupid at first ? Yes, first impression is like it is so. But if we take things like that
square-circle is html object which has CSS property of border-radius equal to 25% - this starts making sense. We get an object which looks like:
If by saying "defined", you have in mind that concept must be defined unambigiously between all people - well then you will have a big problem, because HUGE amount every now and then concepts can be defined differently. And there is no true definition. As long as all conditions are met for the concept at hand - it's a valid definition. We can say only that.
I'd agree with you that "there are no concepts we cannot define", but my agreement is basically a matter of definition. A concept is, by definition, something accessible to human contemplation/intellect, e.g., according to https://www.merriam-webster.com/dictionary/concept it's "something conceived in the mind". So that "in the mind" part is what I'm thinking is crucial.
However, the "define" part of your question maybe itself requires further clarification/definition. If I have a concept in my mind, does that necessarily mean it's "definable" by me? An unambiguous "yes" would be if I can write down a finite sequence of symbols (from a finite alphabet), e.g., an English sentence or even an entire book, whose meaning is that concept. Of course, that presupposes that everybody understands the meanings of my symbols/words/sentences/etc the same way that I do. And for the scope of this answer, I'll just assume everybody understands English identically (or close enough).
Whether or not there exist "humanly-inconceivable things" that maybe some super-advanced outer-space alien "understands" is maybe an open question. For example, in the other direction (where we're the super-advanced ones), I don't think I'll be explaining algebra to my dog anytime soon. Nevertheless, I don't offhand see why we should consider ourselves the be-all and end-all of evolution. Just the spur-of-the-moment top-of-the-line model ... like dinosaurs once were. So, yeah, if I had to guess, I'd guess there are "humanly-inconceivable things". But they're not concepts, per se. That's, by definition, a human thing.
And I guess the one remaining loophole re (humanly-conceivable) concepts, is whether or not I can have a concept in my mind that I can't find words to explain/define. Well, that's what we have poets for.