Which one would you consider correct:
This quick note is to just inform you that...
This is a quick note to just inform you that...
Obviously, the second one is widely used, but I don't know about the first one
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I would suggest:
This is a quick note to inform you [...]
If you would like to maintain the "just", it should go in before the subject:
This is just a quick note [...]
Without the "just" your first sentence would be valid. It focuses on the purpose of the note, whereas the second focuses on the existence of the note: "A note to inform." vs. "This is a note."