The connector on the end of the headset's cable is a P-Series. Specifically it is PN PAG.M0.6NL.AC65NZ. You will need to purchase the strain relief separately. If you are looking for the socket Then you need PN PRG.M0.6NL.LC65N for inline connections or PTG.M0.6NL.LC65N for Panel connections.
Unless you own or want to buy the crimp tool, you will need buy solder contacts (and buy extras if you've never soldered these before.) In my experience, universal crimpers don't work well enough with LEMO parts. If you are doing panel mount consider how important FAA compliance is in your situation and consider paying an avionics guy or consulting a DAR (or whichever governing body and official is salient to you.)
UPDATE: On closer reading of the P series catalog, the first letter of the last sequence determines whether the connectors are crimp or solder. A and L indicate male and female solder, respectively. C and M represent male and female crimp. The part numbers provided above are, therefore all for solder connection.
Note that it is possible Bose had a special variant made for them. In most cases this means special keying to make the connector proprietary. It is possible file keys off of generic connectors but that is not really recommended. I disclaim any responsibility if you waste $23 or have an uncomfortable conversation with your FSDO or DAR. Note also that these parts can be a bit hard to research because they are actually made by Redel, a subsidiary of LEMO. Calling these things LEMO connectors is a bit like calling a Porsche 911 a Volkswagen 911.
I've included links to digikey and mouser in my references. Either site is a great source for parts in the US and should stock all the parts I've referenced.
Additionally, here is the P-series catalog.