I've an arduino nano 33 iot and I'm working with the first time with bluetooth. So far BLE is working fine, I can use an app on my smartphone to check the services and characteristics. Is it possible to send data over serial via Bluetooth on this board ?
This board has multiple processors on it. The main processor (SAMD) rus your arduino code. There is also an ESP32 based processor which runs the WIFI and the Bluetooth. There is a pretty featured library for controlling the WIFI called WIFININA. This library does NOT have any bluetooth functions though. Therefore, until someone writes a library which lets you control the bluetooth functions of that ESP32 chip, I don't think you'll be able to send arbitrary serial data via bluetooth. It is "possible" to do this but not without a lot of technical wizardry.
Here are some links to examples of what the code might look like if you were writing code directly to the ESP32 chip here https://github.com/espressif/arduino-esp32/blob/master/libraries/BluetoothSerial/examples/SerialToSerialBT/SerialToSerialBT.ino