Yes, it will run full speed. But you also will need to make the right connections, likely needing some custom wiring soldering. I expect the connector on the end of the two wires from the fan motor is not the common 3-hole or 4-hole female fan connector that would fit onto the pins of the mobo header. However, it might fit anyway as long as you make sure to connect the right way around. Whether this works, or whether you have to custom fit some other way, here's how.
On the Mobo HEADER with 4 pins there is a plastic "tongue" sticking up beside the first three pins - Pin #4 is beyond the tongue. Pin #1 is the Ground pin, and Pin #2 is the +12 VDC power supply. On the FAN, most commonly the BLACK wire is for Ground, and the RED wire is +12 VDC. Just connect that way and it will work. NOTE one thing. The mobo header normally checks the fan's speed signal to ensure that it is running, and will send out warning messages if it fails. Your fan has no speed signal wire to header Pin #3 so it cannot send that signal to the mobo. Thus the header will tell you the fan has failed. In BIOS Setup, see if you can disable the warning messages from that header to stop the annoyance.
FURTHER, how do you plan to connect to the mobo header? Do you not have another fan plugged in there already? If you do, are you going to use a Splitter to connect two fans to the header? IF you do that, don't worry about the speed signal problem. A Splitter will only send to its mobo header the speed of ONE of its fans, and the output connectors of the Splitter are different for this reason. ONE will be identified as the only output that sends back its fan's speed signal. Just ensure that the REGULAR fan is plugged into that output, and the new fan into a different one.