I have submitted my paper to Proceedings of the AMS one month ago and we haven't gotten a response yet. Can I upload my paper to arxiv despite of this? Would there be any conflict? About accepted ones, they have written that there is no conflict and so: Can I deduce from this that there wouldn't be any conflict for submitted ones as well?
Yes, in mathematics you can absolutely upload both submitted and accepted papers. I am not aware of a single reputable mathematics journal that would take issue with this. (If it helps, within the last few months I uploaded a paper to the arxiv after I submitted it but before acceptance.) What I do not recommend is to upload to the arxiv papers which have been formatted in the journal's style file or which incorporate copyediting done after acceptance by the staff of the journal.
If a journal is published by a big academic publisher and you are the kind of person to worry about this at all (honestly I didn't, for many years), you should be able to find online a relevant policy for uploading preprints. In all of my experience, even big publishers like Elsevier are quite reasonable about this...because they have to be in order to do business with mathematicians. The Proceedings of the AMS is published of course by the American Mathematical Society. The current managing editor of this journal posts his preprints to the arxiv. The linked-to preprint was submitted to the arxiv and for publication on the same day; a modified version was uploaded to the arxiv after the paper was accepted and before the paper was published. So I think you have nothing to worry about here.