Can i claim damages from neighbor if a coconut tree in their property fell on my property?

Yes,you can claim damages by filing civil suit against neighbor. 

Yes, you can.....................................

If the neigbor's tree fell on your property due to storm or any other natural calamity then you can inform the neighbor about the incident and can give him an estimation of the damages caused due to this and ask him to pay the same within a time stipulated thereby in your notice.

If he fails to comply with the demands made, you can sue him for recovery of damages or compensation

Yes, you can claim damages from your neighbor if a coconut tree on their property fell over and damaged construction on your property. 
 

You can Claim Damages under 
Negligence:
Your neighbor may be held accountable for negligence if the tree was in a potentially dangerous state and they were aware of about it 

Nuisance:
If the tree posed a hazard or interfered with your enjoyment of your property,

Very well analysed, opined and advised by learned experts, I endorse.