What I currently have for this is probably a literal translation:
In amor cum haec lingua cado
Thank you 🙏
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Latin has an inchoative suffix -sc- which indicates that a certain state is beginning, and which is quite productive (rubesco, senesco, reconvalesco, ...). And indeed, there is the verb amasco – "to begin to love", so that you could say
hanc linguam amasco
Oxford [English-to-Latin section (under "fall")], offers "adamo" = "to fall in love with", taking the accusative case. (In the Latin-to-English the definition of "adamo" is "to love passionately". I am always suspicious when the two sections fail to coincide.)
Lewis & Short gives "to love truly, earnestly, deeply". Cicero used this verb only in the perfect & pluperfect tenses; while Quintilian (2.5.22), used the present tense.
"linguam hanc adamo." =
"I am falling in love with this language."/ "I deeply love this language."
The simplest way may be to use "capio" = "to captivate" and the figurative, "to enthral" (Oxford). Therefore, using the passive, "capior":
"lingua hac capior." = "I am enthralled by this language."
Thanks to Sebastian Koppehel (CHAT) for:
"amore capi vel incendi" =
"by love to consumed or inflamed".
Using the passive of "incendo":
"hac lingua incendor." = "I am inflamed (with passion) for this language.".
Seb, himself, thought that this might be over-the-top for the love of a language, as opposed to the romantic love of a person; still, it provides a poetic slant.
Although the inchoative suffix -sc- is productive, I would advise against using it in everyday speech unless the verb is already a common one. I could say "pugnasco" (unattested) or "puellasco" (a couple usages), but it would call attention to itself in a way that doesn't seem fitting for the phrase "falling in love." Since I can only find one example of amasco in Naevius (~200 BC) and one example in Diomedes (AD ~375), it's a pretty good guess that it wasn't commonly used.
Latin doesn't use the idiom "I fall in love," but it uses a rough equivalent all the time: "I have begun to love," i.e. coepi amare. Here are a few examples:
amare valide coepi hinc meretricem (Plautus, Mercator)
- Incidentally, the previous lines of this speech have two other, more colorful equivalents of falling in love: amorem Venus mi hoc legavit die and animus studio amotus puerilist meus)
eius filiam ille amare coepit perdite (Terence, Heauton Timorumenos)
- Terence has a few more examples, including amare coepi alone.
Porcus est, quem amare coepi (Pomponius, Atellanae)
Keep in mind, though, that amare need not refer to romantic love:
Nunc venio ad Brutum, quem ego omni studio te auctore sum complexus, quem etiam amare coeperam (Cicero, Epistulae ad Atticum)
Though coepi, -isse is perfect (contrary to my earlier assertion), this seems to capture the sense of English "falling in love," i.e. it is a process that has already begun. There are a few scattered examples of incipio + amare, which almost all seem to occur in the subjunctive. coepi + amare is more common in the corpus.
Here's my suggested translation:
hanc linguam amare coepi.
You're free to add adverbs as in the above examples (valide = deeply; perdite = desperately) to make your meaning clearer.
Others have suggested many ways to express falling in love in Latin. Let me address the grammar of your suggestion, even though it was too literal as a translation.
There is only one problem, but it occurs twice. Pay attention to which case is needed with each preposition:
Cum requires the ablative. Both haec and lingua need to be in this case: hac lingua.
In requires the accusative or the ablative. Accusative is used for movement ("into") and ablative for location ("inside"). In this situation movement is intended, so you need the accusative.
Thus you'd get: In amorem cum hac lingua cado. But this does not mean "I fall in love with this language", at least not in the usual sense. It is perhaps better translated as "I am in the company of this language and I sink down into love". A bit too melodramatic, I'd say.
A less literal (but perhaps more idiomatic) proposal:
Amore huius linguae accendor (“I am ablaze with love for this language!”).
Or, slightly less hot perhaps:
Studium eius linguae me excitat (“The interest for [love of] this language excites me,” i.e. “I am very much interested in, I love this language”).