Generally, you need to use two SFPs that are compatible with the same Ethernet variant (speed, wavelength, fiber type). Both the J4859D and the MGBLX1 are running 1000BASE-LX(10), so that's not your problem.
Please check the switches' logs to see if they've accepted their respective transceiver.
allow-unsupported-transceiver is only necessary on the 2930 for non-HP transceivers (=MGBLX1), the J4859D should work without. When an inserted transceiver is accepted, the port LED lights up for appr. one second. If it's not accepted, the LED keeps blinking.
If each switch has accepted its transceiver, the only thing left is to use the correct cable. 1000BASE-LX runs over single-mode fiber. Just for completeness, fiber cables are generally crossed, ie. the left-hand transmitter needs to connect to the partner's right-hand receiver and vice versa.
Some long-range transceivers may require an attenuator on short links to avoid blinding the receiver, but that is not a problem here.
It is important to note whether the link LEDs light up on cable insertion (=layer 1 up). Equally important are log events on transceiver and cable insertion.
If everything fails, you can test a single transceiver by having it link to itself with a simple loopback cable (=simplex cable). Most duplex cables can be split to be used as two simplex cables. Of course, you need to make sure that that doesn't cause a network loop - configure a spanning tree protocol or use only an otherwise empty VLAN on the link.
And of course, any configuration on the switch interfaces or VLANs need to match - by default, they should talk to each other. To check basic layer-2 connectivity you can display the MAC table (for the respective VLAN) on the connected port.
The Cisco Business 350 series seems to be managed, so you should check their config as well.