As an academic myself, I can tell you that to gain reputation in academic circles, two things are needed. First, you must be able to demonstrate expertise in the relevant fields. Second, you must convince the people involved that you are worth having around and that you are a good fit for various positions, an activity which falls under the umbrella of 'networking'.
(I provide an answer here that is somewhat broader than the specific scenario of 'knowledge of other cultures' presented in the body of the question, to cover similar situations for future readers.)
Demonstrating your expertise in an academic field is definitely an Intelligence check of some description. The exact skill associated with the check depends on what kind of academics you are trying to impress. Looking through the Intelligence-based skills, I find the following relevant skills.
Intelligence (Arcana) will impress wizards and researchers of magic.
Intelligence (History) will impress historians, geographers (of the human society type), archaeologists and politicians. In the specific scenario presented in the question (knowledge of other cultures), this appears to be the most relevant.
Intelligence (Medicine) will impress doctors and physicians.
Intelligence (Nature) will impress zoologists, geographers (of the natural environment type) and botanists.
Intelligence (Religion) will impress theologians and clergy. If the 'knowledge of other cultures' in your particular scenario is religious in nature, then this skill would apply.
(I omit Intelligence (Investigation), because that represents your ability to find information, rather than any particular knowledge base.)
You may also consider making an Intelligence check with a non-Intelligence skill. If your target audience are more arts than science, and are impressed by flowery words and well-constructed essays rather than specific knowledge, then Intelligence (Persuasion) might be an appropriate check to write impressive essays. If you want to impress some zoologists but don't have Nature proficiency, then Intelligence (Animal Handling) may work instead.
If you want to impress a specific group not covered by these fields, we need to decide whether proficiency applies to the check.
Proficiency in a particular set of tools may be relevant. Proficiency in navigator's tools or cartographer's tools would probably let you add your proficiency bonus to an Intelligence check to impress geographers, depending on the circumstances and how 'applied' as opposed to 'theoretical' these academics are.
But not every academic trade would have a set of corresponding tools. For concreteness, suppose you wish to impress some mathematicians. There is no Intelligence (Mathematics) field in D&D 5e, and neither is there a set of Mathematician's Tools, but hope is not lost. One might argue that proficiency in a gaming set gives you the numerical background to talk mathematics. Or you might argue that your studies of the arcane also covered mathematics, allowing you to add proficiency from Arcana.
If neither of your existing proficiencies match, consider the Skill Variants in DMG p.264 which suggest using Backgrounds or Personality Traits to determine proficiency rather than a fixed set of skills. Even if you are using the standard Skill rules, Backgrounds and Personality Traits are still a good match for determining proficiency for checks which do not fall under any particular Skill, or which are highly specialised. For instance, in my current game, I have a barbarian who has a background as a farmer. Although she lacks proficiency in the Nature skill, I would allow her to take proficiency on checks specifically related to agriculture and farming; such cases are sufficiently specialised to not be unbalanced. In this mathematics example, if your character has the Sage background or similar and has studied mathematics in their back-story, then it would be reasonable for them to add their proficiency bonus to impressing mathematicians.
If, after all this, you still don't have proficiency in impressing the academic circle you want to impress, then you can always make an Intelligence check without proficiency. Being a bard, you still get to add half your proficiency bonus anyway due to your Jack of All Trades feature.
I would not allow a check with proficiency to impress generic academic expertise upon an audience, because there is no such thing as generic academic expertise. At best, you could make a straight Intelligence check (affected by Jack of All Trades, of course), although I would advise against this. For optimum results, pick a particular field (or fields) you want to specialise in, and focus your efforts there. Hypothetically, you could target all fields individually, but I would make reputation in each field separate.
Academic expertise is a necessary but not sufficient condition to increase your reputation. The adage "It's not what you know, but who you know" is just as true in academic circles as it is elsewhere.
To go up in reputation, get positions of tenure and receive invitations to future conferences, people need to like you and know you. Achieving this in D&D will involve Charisma checks.
There are several contexts in which Charisma checks are appropriate. If you are presenting a talk, that is a Charisma check. Charisma (Performance) is the skill most suited to giving a speech for an audience, but Charisma (Persuasion) could also be used.
Your character would also be chatting and socialising with the academics before and after any presentations. This fits under the Carousing rules mentioned by indigochild, and requires the use of Charisma (Persuasion).
Bringing it together
If I were to run this, I would request an appropriate Intelligence check to ensure that you had the correct knowledge and academic expertise, followed by a Charisma check or two to impress the academics. I might apply a bonus to or adjust the DC of the Charisma checks depending on the outcome of the Intelligence check.
If the choice of skill to use for a particular check is not obvious, I would ask the player to persuade me as the GM which skill should be used and whether proficiency applies. This can see some creative uses of skills and takes some of the burden off the GM.
You can also consider using the Renown rules (DMG p.22) to encapsulate the effects and progress of this character within the academic spheres. They can make the above checks during downtime to increase renown. And if the character finds particularly interesting lore during an adventure, then they can gain renown automatically by sharing that lore with those scribes.
If you choose not to use renown, then having interesting or valuable lore obtained on the adventures can give advantage on one set of checks to increase standing with the academics.