<p>My FDM printer bed moves on the Y-axis and the print head moves on the X-axis and raises on the Z-axis. When printing rectangular objects (a model of Notre Dame in this case), are there print detail quality advantages to aligning the model perpendicular to the X or Y axis, or at 45 degrees? Part strength is not an issue and support is not needed.</p>

<p>Thanks.</p>

In short: Not really.
longer version: It depends.
The main culprit of losing details in this case would be the weight and speed of the thing moving. So if you have a heavy X-axis carriage, acceleration and decelerating the carriage won't be instant. Same with the bed (Y-axis).
Another culprit can be slop in the system, so check your linear bearings and belt tension.
Also keep in mind that you are printing on the bed, so the weight of the Y-axis increases while the print progresses. This shouldn't be a problem for small prints, but if your print becomes bigger it can decrease the quality. Another factor is that every print will bend a little the higher it gets, so if you print a tall slender object, don't accelerate the bed too fast ;)
To summarize, for high detailed prints:
Lower the speed
Check the system for slop (tighten belts, and align linear bearings)
Take the lightest axis for the highest detail (keep the weight of the print in mind)
One thing that you can do to test your machine is to test the ghosting on each axis (
https://www.thingiverse.com/thing:277394
).