Hi
I think the question could be answered more precisely if the HP product numbers are included. Sometimes the product names are misleading.
Assuming that the HP 1910-16G Switch has product number JE005A and the HP X122 1G SFP LC BX-D Transceiver has J9142B based on the HP documentation the answer would be no.
Very useful document are the quick specs
http://h18006.www1.hp.com/products/quickspecs/13677_div/13677_div.HTML
When you go to Accessoires you can find a list of the supported transceivers
HP 1910 Switch Series accessories
Transceivers
J4858C HP X121 1G SFP LC SX Transceiver
J4859C HP X121 1G SFP LC LX Transceiver
J8177C HP X121 1G SFP RJ45 T Transceiver
JD118BHP X120 1G SFP LC SX Transceiver
JD119BHP X120 1G SFP LC LX Transceiver
JD493AHP X124 1G SFP LC SX Transceiver
JD494A HP X124 1G SFP LC LX Transceiver
JD089BHP X120 1G SFP RJ45 T Transceiver
HP X122 1G SFP LC BX-D Transceiver J9142B is not listed here. So it means that it is still not supported on this switch.
What you should know about the BX transceivers is that they are used in pairs. A BX-D (Downstream) should be connected to a BX-U(Upstream). D to D or U to U will not work. The part that can be used with this transceiver is HP X122 1G SFP LC BX-U Transceiver (J9143B)