Can I add Parmesan to a béchamel sauce and have a Alfredo-esque sauce?

Pretty self explanatory. I got an Italian cook book (Essentials of Classic Italian Cooking by Marcella Hazan) but it doesn’t really elaborate on an Alfredo sauce from what I’ve seen. Just making sure that this won’t be tarnishing some multi century long rule haha. Also would appreciate semi quick answers, my dinner depends on this

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from AskCulinary https://www.reddit.com/r/AskCulinary/comments/n27ylv/can_i_add_parmesan_to_a_béchamel_sauce_and_have_a/