How do I create a conference concept document?

Try an infographic but if budget/skillset doesn't allow use a word doc and create a pdf. Whatever you do keep it short and sweet and a better suggestion is to use the body of an email for the actual pitch and the attachment/one pager or close too with supplemental info. No one likes reading and this is just an opener so they get a feel - you can answer detailed questions in a call/meeting

In my experience it's tricky. I've been burned a few times by the "we totally love it, we're behind you" thing, until you actually need money :)

I'd say a document like that needs to address the following.
Audience (Community the event will serve)
Competition (which other events are out there, maybe a count, maybe their primary location, or size, etc)
If you can figure it out, demographic if likely attendee (I don't know this until after my events or at least after tickets start selling, but you might be able to do a survey)
I think lastly, what's in it for the sponsor? Either your full sponsorship options list, or a description, likely size of attendee pool, email blasts, etc.

Off the top of my head that's what I'd put into something like that, if i was making one

Here is a good example which I feel might be extremely helpful for you: https://www.civicus.org/images/65th_UN_DPI_NGO_Concept_Note-FINAL.pdf
Besides if you do have any questions give me a call: https://clarity.fm/joy-brotonath