Based on the fact that the question is in a case where there is no contract, I am assuming that this includes that there is no overriding minhag ha’ir that can determine what the default intention is.
Such a scenario is logically similar to a case in which the renter and the landlord didn’t discuss if the rent is yearly or monthly, and then the year was turned into a leap year (back when it wasn’t based on a pre calculated calendar). This case is the same thing, since in essence the only difference is whether or not the renter has to pay for the extra Hebrew month. (Although perhaps there is room to debate this logic, the reasoning behind the Halacha would seemingly apply in both cases regardless).
The Halacha in that case is that the house is in the chazaka of the landlord, and the renter needs to prove that he has a right to the extra month. The Rambam in Sechirus 7:2 says;
הֲרֵי חֹדֶשׁ הָעִבּוּר שֶׁל מַשְׂכִּיר שֶׁהַקַּרְקַע בְּחֶזְקַת בְּעָלִים וְאֵין מוֹצִיאִין דָּבָר מִיַּד בַּעַל הַקַּרְקַע אֶלָּא בִּרְאָיָה בְּרוּרָה. וְכֵן בַּעַל הַבַּיִת שֶׁאָמַר לִזְמַן זֶה הִשְׂכַּרְתִּי לְךָ וְהַשּׂוֹכֵר אוֹמֵר לֹא שָׂכַרְתִּי אֶלָּא סְתָם אוֹ לִזְמַן אָרֹךְ עַל הַשּׂוֹכֵר לְהָבִיא רְאָיָה וְאִם לֹא הֵבִיא בַּעַל הַבַּיִת נִשְׁבָּע הֶסֵּת וּמוֹצִיאוֹ מִן הַבַּיִת:
In this case it would seem that the same Halacha would apply, the landlord is assumed to be the muchzak on the extra month and the renter would be the one that has to prove otherwise. (Perhaps, if he always paid on the first of the secular month that can prove that the agreement was secular months, but if the landlord complained right away to use the Hebrew months then seemingly he’s in the right)