When it comes to time-delay supplemental fuses it can be a little tricky. This is because the fuses have to meet certain performance points as detailed by UL 248. However, outside of those points, fuses from different manufacturers can vary, especially with regards to inrush characteristics.
Because of this some-manufacturers may have more time-delay for inrush considerations than others. This may even change based upon the ampere rating of the fuse and inrush point (or current value) and time duration of inrush spike. This may be the case here and is can be more of an issue for small control transformers because they can have an unusually high inrush.
However, a good comment was made by the previous person...since the FLQ opened maybe there was a problem with the transformer.
In this case, I would probably recommend using the FLQ for this application unless the FNQ can be sized a little higher.
Keep in mind, other applications with the FNQ may not have this issue, it can depend upon the control transformer used.