I would like to know if the following is possible using a Nano 33 BLE or and Arduion that has built in bluetooth.
I have a telescope with a Firefly serial to bluetooth adapter connected to it for communications. Current I have a PC and other devices that can connect to the telescope using the firefly serial to bluetooth adapter. To do this the software that connects need to first pair with the adapter. Can this be done with an Arduino also?
Yes, you shouldn't have any issue pairing an Arduino with Bluetooth to generally any Bluetooth enabled device.
The question is, why the Arduino? I've very interested to know what you'll be doing with the pair?
Bluetooth Classic and BLE are two incompatible protocols that only share the name and the frequency band (2.4GHz). The Arduino Nano 33 BLE is a BLE only device. It does not support Bluetooth Classic.
I suspect your Firefly uses Bluetooth Classic with the Serial Port Profile. BLE does not support SPP because it is not energy friendly and a bit outdated.
Pairing is an optional feature in BLE. Paring is usually done through the mobile operating system and requires the user to confirm the process via some pin code. The ArduinoBLE library does not support pairing. BLE devices can just connect. Connecting can be done from within the app.
There are modules that support Serial over Bluetooth Classic with little configuration. You should investigate which module would be compatible with your adapter. There are a few people in the forum that have knowledge about these modules. You may want to provide some more detailed information about the Firefly adapter to help them help you. e.g., link to datasheet or other useful information.
It is cheaper than the Arduino Nano 33 BLE, also supports WiFi (not at the same time) and the antenna area is better protected. The Nano 33 IoT is not as powerful as the Nano 33 BLE, but only when you want to run machine learning stuff. It sounds like this does not matter in your case.