Not easily, but it's possible.
If both devices are MIDIbox based, it might be easier to get one to send LCD commands through the other (maybe via CAN bus, or I2C or even MIDI).
Having them share the LCD bus is difficult to get right. Both devices would have to correctly handle when access to the LCD is given/taken, again if this was MIDIbox for both, they'd have to switch their ports to hi-Z when they don't have access... so there'd be a lot of coding to do... well, I could go on in great detail how I'd do it but I'll just give the short answer - it's possible, but there are easier ways to achieve the same thing.
If they are not both MIDIbox devices (i.e. you can't write code for both things you want to share the LCD), then it isn't possible - each device would assume it has sole access and wouldn't play nice, wouldn't refresh when switched to, etc.