No, bizarre would not really fit here. The example you proposed would be translated as “I have an odd question” or literally, “I have a bizarre question”.

Also to avoid here : “une question aléatoire” (= randomly chosen from a given set of questions) or “une question hasardeuse” (= dangerous in some way, sometimes figuratively, i.e. hard to solve rationally, or which could lead to some trouble, like people quarelling)

There’s no literal translation, but to convey this meaning of the word random (= “nonspecific / arbitrary / undirected”), you could try one of these :

J’ai une question [quelconque / générale].
J’ai une question [sans rapport / qui n’a rien à voir]. (implied : “… avec le contexte / le sujet”)

The second one actually has a slightly different meaning (better translated as unrelated question). I’ll leave it here for now because it’s close enough and can be a good variant in some contexts.

And then you also have some “casual context only” variants :

J’ai une question au pif.
J’ai une question comme ça.
J’ai une petite question.
J’ai juste une question.
J’ai quelques questions en vrac. (the plural form was more natural here)