I don't think these colleagues are on the selection committee of the internship program
If they are not, then personally I would not send an email to them about the student, not necessarily out of ethical considerations, but because I would expect sending an email to make no difference to the student's outcome.
If I were on such a committee, and a colleague not on the committee approached me to ask that I give different treatment to one applicant whom they know of via a professional contact, I would be inclined to ignore that request completely. I would expect that if their professional contact has anything relevant to say about the applicant, it should be said in their application, especially if the professional contact is the one who wrote the reference letter. So even if your email does result in your contact approaching somebody on the committee about your student, I doubt it will have any consequences for your student.
Now consider it from the contact's perspective. If I got an email from a professional contact asking me to make sure that a committee in my institution gave careful consideration to one particular applicant who is their student, and I was not on that committee, I would respond with a polite email saying something like: "Hello Professor X, hope you're well. I am not a member of the committee so I will not be involved in considering applications; if there is something you left out of your reference letter then you could try contacting the committee to send an updated reference letter. Best wishes to your student." I would not approach anyone on the committee about it.
On the other hand, there is a risk that your contact will think you are asking for some unfair advantage for your student, because an email asking for an unfair advantage looks exactly the same as an email that might be considered a legitimate request. A nepotist would never outright say "please lobby the committee to accept my student", they would write the same email you would plan to write, because that's plausibly deniable and if the contact is on board with it then they will read between the lines. So I am not sure there is a way of wording such an email which, from the recipient's perspective, totally eliminates the idea that you might be asking for nepotism.
Of course, you may know your contact well enough to know they would not think that, in which case there is probably no downside to sending the email, but I would still expect there to be no upside either.