First and above all: congratulations on your paper being accepted! Let's look at the half-full glass: it is such a relevant progress in your field that one peer of you found it so relevant that he decided to work on the same topic/approach on its own.
Regarding your question, first the malicious thinking, that your peer stole the idea from you. From an external observer, both of you are innocent until proven guilty, but the same facts you expose here can be easily turned the other way around.
If I were your malicious colleague, I would claim that I planned and submitted the work to the conference as soon as it was announced (way before conferences submission deadline, usually even minor conferences are announced one year before), and that I had informal discussions with you during our day-to-day interaction and that you stole the idea from me.
How to protect yourself?
You have a solid trace of your work, which is the mail discussing the work with your co-authors dating before the discussion with your colleague. The colleague of yours can claim they started working on that idea many months before your discussion, and either one of the following:
- the idea sharing with you never took place;
- he presented the idea to you and you stole the idea from him.
To protect yourself from such an occurence, you should signal to your co-authors that unfortunately someone from your group took inspiration on the main idea of your paper, because you discussed informally with them during day-to-day interactions and presented it at a minor conference. Forward them the contribution to the conference of your colleagues and give them an estimate of the time schedule. Plagiarism is (luckily) taken more and more seriously, better to prepare all of you from such an occurence.
Please also signal the situation to your PhD advisor (in person, not via email, this is a confidential topic and you do not want to put yourself in trouble because a misunderstood sentence or an ambiguous sentence that you wrote). You must have had some internal presentation and discussion on the topic before submitting the paper, before discussing the topic, has your fellow PhD similar records of his interest/involvement on the topic?
I unfortunately foresee the chances of your PhD advisor thinking it would be a good idea you start to work with your PhD peer on a follow-up paper on the topic: prepare yourself to defend your ground, stating that such a thing must be obviously discussed with the other co-authors.
However, I would not discount the possibility that your peer was stuck in a rut, thinking about problems to be solved in your field and he found the discussion with you so inspirational that he started using your approach, discussing it with other colleagues in your department (who are his co-authors?) and by affinity in sources, he ended up with a very similar research process and conclusions to the one you reached, along the process forgetting the casual discussion he had with you.
In short: I unfortunately see no good outcomes from this situation. Try to avoid as much as possible interactions with this PhD student.