Hi,
A Brother MFC-J1010DW is listed in About AirPrint - Apple Support. Is this your exact printer model?
If so, the printer supports Apple's AirPrint technology, and you should be able to use it without additional software/apps, provided that the printer and the iPhone are connected to the same network (Wi-Fi router). Connect via the router (not directly from the iPhone to the printer).
Use AirPrint to print from your iPhone or iPad - Apple Support
However, for some reason, information on the following Brother support page indicates that Brother Mobile Connect is needed.
https://support.brother.com/g/b/downloadlist.aspx?c=us&lang=en&prod=mfcj1010dw_us_eu_as&os=112
If really necessary, you could perhaps test that iOS app as well.
https://apps.apple.com/us/app/brother-mobile-connect/id1516235668
You may also want to check that the latest printer firmware is installed.
In case of difficulties recognising an AirPrint printer, you could try a power cycle sequence (do not forget the router):
Power off the Wi-Fi router, the printer, the iPhone, and all Macs/iPads/other computers and devices. Wait two to three minutes. Power on the router. Wait two to three minutes. Power on the printer. Wait a minute or two. Power on the iPhone. Finally, power on any Macs/iPads/other computers and devices.