No, and that answer holds regardless of the relationship or the reason for asking.
Inmates retain privacy rights over their personal communications from a civilian standpoint. The facility monitors calls and mail as a matter of institutional policy, and that is disclosed to everyone using the system. But that monitoring belongs to the institution, not to family members, fiancees, or anyone else on the outside. There is no mechanism through which a third party can request or access an inmate's call logs, mail records, or visitor history.
This applies equally to spouses, parents, and anyone else with a close personal relationship to the inmate. Being the person funding the phone account or sending the letters does not create any right to see how those privileges are being used with others.
The harder reality underneath this question is usually about trust rather than access. If the concern is about who your fiance is communicating with, no record is going to resolve that in any satisfying way. People who want to hide something find ways to do it, and discovering a call log would raise more questions than it answers.
What you do have control over is what you fund and what you participate in. InmateAid's discount phone number is tied to one number only, meaning calls made through that number reach you exclusively. That does not tell you what is happening on the facility's general phone system, but it ensures your money is going toward calls to you.
If the trust concern is significant enough that you are looking for ways to verify his communications, that concern itself is worth examining directly with him.