Can I ground a ceiling fan to another electrical ceiling box nearby?
Yes you can

Recent NEC editions allow you to run a retrofit ground wire (equipment grounding conductor) from an ungrounded box/circuit to a grounded branch circuit's ground wire, provided the donor circuit has a large enough ground wire to be a suitable ground for the circuit being retrofitted.  (For instance, if you are grounding a 20A circuit this way, the donor circuit would need to have a 12AWG or larger ground wire.)  The permission for this is given in NEC 250.130(C):


  (C) Nongrounding Receptacle Replacement or Branch Circuit
  Extensions. The equipment grounding conductor of a
  grounding-type receptacle or a branch-circuit extension shall
  be permitted to be connected to any of the following:
  
  (1) Any accessible point on the grounding electrode system
  as described in 250.50
  
  (2) Any accessible point on the grounding electrode conductor
  
  (3) The equipment grounding terminal bar within the enclosure where the branch circuit for the receptacle or
  branch circuit originates
  
  (4) An equipment grounding conductor that is part of
  another branch circuit that originates from the enclosure
  where the branch circuit for the receptacle or branch
  circuit originates
  
  (5) For grounded systems, the grounded service conductor
  within the service equipment enclosure
  
  (6) For ungrounded systems, the grounding terminal bar
  within the service equipment enclosure
  
  Informational Note: See 406.4(D) for the use of a ground-fault
  circuit-interrupting type of receptacle.


Furthermore, since ground wires do not need to run with the circuit they are grounding as per NEC 300.4(B)(2):


  (2) Grounding and Bonding Conductors. Equipment ground‐
  ing conductors shall be permitted to be installed outside a race‐
  way or cable assembly where in accordance with the provisions
  of 250.130(C) for certain existing installations or in accordance
  with 250.134(B), Exception No. 2, for dc circuits. Equipment
  bonding conductors shall be permitted to be installed on the
  outside of raceways in accordance with 250.102(E).


, you can simply run a bare 14AWG wire from the new ceiling fan box, through the ceiling, to the "donor" box, bring the wire into the box, and connect it to the existing ground wire bundle there.