I gather this is an expensive, quality fixture that uses dual T5 bulbs, and it has an electronic rapid start or progammed start ballast (that is to say, there's a small time delay before it comes on).
All fluorescents are discharge lights (like neon, mercury vapor, sodium and metal halide). They all work by putting high voltage between the ends of the tube, which "strikes an arc" down the length of the tube, and the arc makes the light.
These rapid or programmed start ballasts have a second trick, they also put a low voltage between the two pins on each end of the tube, which preheats filaments in the ends of the tube. That is the orange glow you see. Not every ballast does that, but these do, and those warm filaments mean the arc can strike at a lower voltage and with less wear-and-tear on the bulb.
As a bulb ages, it takes higher and higher voltage to strike the arc initially. (Also, you're getting less light out of it.) When the voltage goes so high that the ballast can't start the bulb anymore, we call the bulb "dead".
So if you look closely during the first second of startup, you should see that orange glow on BOTH ends of the bulb. If you don't, the bulb is not seated properly in its socket, or the filament is burned out. If you do, the bulb has probably reached the end of its life. The new electronic ballasts turn off the filaments once the arc has struck.
Another possibility is the ballast has burned out. That is the problem with these electronic ballasts, often made in China with RoHS lead-free solder in which the zinc crystallizes out, and capacitors which eventually fail (especially if they were made during the 2005 capacitor plague, google it).
It doesn't have starters - those are a throwback to the 1950s.
I would start by removing the bulb(s) and putting them back in. Just to see if a plug is not fully seated. Look for damaged lamp sockets while you're at it.
Most likely, it's the bulb.
If that's not it, it must be the ballast. A fixture of this quality is designed to be serviceable and replace sockets and ballasts. While you're in there, look for loose wires before ordering another ballast.
Before buying a ballast, also price LED "fluorescent tubes" which require no ballast, or replacing the whole fixture with an LED.