Relativisation and limiting the scope by introducing a temporal, spatial, logical or value-based limitation can generally be interpreted as an attempt to "soften" a demand, and thus to make a question more polite. This is because it leaves a bigger space for the asked person to react. This is an explanation of the fact that the use of Konjunktiv (subjunctive) is considered more polite than the use of plain Indikativ. Adding the qualifier gern is an additional limitation of the demand, also raising associations which stem from ordering something. The following sentences are an ordered list of demanding the same, ordered by their degree of politeness:
Ich habe eine Frage.
Ich hätte eine Frage. [Using Konjunktiv as means of relativisation]
Ich hätte mal eine Frage. [Adding "mal" as temporal limitation]
Ich hätte gerne mal eine Frage. [Adding. "gerne" as a value-based limitation]
Ich hätte vielleicht gerne mal eine Frage. [Adding "vielleicht" as logical limitation]
Ich hätte da vielleicht gerne mal eine Frage. [Adding "da" as spatial limitation]
Ich hätte da unter Umständen vielleicht gerne mal eine Frage. [Adding "unter Umständen" as further logical limitation]
This level of politeness can be seen as an exaggeration, and hence it could be meant or taken as ironic. But I would not say, that the phrase is ironic per se.
One could argue that adding gerne changes the "illocutionary force" of the question, i.e. it tells a different intention of the speaker, because by uttering
Ich hätte eine Frage.
the speaker is expressing to have a question, while by
Ich hätte gern eine Frage.
the speaker is expressing the wish to have a question, if you take it literally (I wish I only had a question). Logically this is correct, but since uttering the wish to have a question is clearly not meant, if the question is following directly after, I would say that the cooperative principle directs to the inference, that actually I wish I would get an answer is the intention of the speaker to utter this sentence.
Making the sentence more and more complex, though, will increase the risk of not being taken serious anymore. Speaking in a very stilted manner can also be a means of ironic speech or create a comic impression. Whether one would interpret the sentence given in the question as intentionally or unintentionally comic or ironic, depends on situation and context.