An Amp-Hour is the same as a Coulomb. It is the amount of charge something has.
For example, a battery can "hold" so many electrons. A 1 Amp-Hour battery contains 3600C. A Coulomb is approximately 6*10^18 electrons.
You can think of the amp-hour as the "size" of the container. The larger the size the more electrons there are for use in doing useful things.
It makes no sense to have a fuse rated in terms of Amp-Hours since the only way to blow the fuse would be to push all those electrons in it. But things burn up not due to how many electrons total but how many at localized instant.
For example, you can run a small 5V computer fan at 0.25A almost indefinitely. It may use 1000000000000000C's. But you can burn the fan up by just passing 1C through it if you do it fast enough. (coulumbs/s = A)
Fuses work by burning up when so many amps pass through them and have no "memory"(for the most part). So we could pass 100000000000000C through a fuse without any problem is we do it over a long enough time frame... but 1C could easily blow most fuses.
If 1 Amp is flowing through a fuse then that is 1C in one second. So a small 0.25A fast fuse will probably blow. If we only allow 0.1Amps to flow through the fuse then in one month the fuse would have passed about 250kC.
When dealing with columbs you are generally talking about how many electrons. Amps both how many and how fast. You can have a lot of electrons moving very slow which could give a low current or a few moving very fast which can give a high current.
99.99% of the time when you are worried about something burning up you are dealing with amps and voltage. Fuses should be rated in terms of Watts which is the unit of power and related to temperature. But since fuses generally have a fixed resistance and P = I^2 R we can simplify and talk about amps instead(in this case since R is constant and I = sqrt(P)/R).
To directly answer your question there is not much of a directly relation between amps and amp-hours. They are no interchangeable. One deals with charge/second and the other with charge. It is analogous to speed and distance. They are related but not interchangeable.
I doubt you will ever find a fuse rated in terms of AH's. Chances are you read it wrong or there was a quirky symbol on the fuse(some fuses have weird symbols as they are "Codes" rather than units).