Is transaction malleability the only consensus issue that needs fix to allow lightning network? Is transaction malleability the only thing needs fix in the bitcoin consensus code to enable lightning network to be implemented?

Of course, lightning network itself would need to be implemented, but I'm wondering if after fixing transaction malleability, any changes to bitcoin itself is required.

In particular, if BIP 62 is implemented, would that be enough?
BIP62 has been scrapped and will never be implemented. Instead a much better solution with many more advantages, Seggregated Wittness, has been developed and is currently being rolled out. Its finished code is in 0.13, but not activated yet on main net, that will happen in 0.13.1.

As far as I know, yes, this is the last thing required to enable LN. From what I've seen LN was actually possible without malleability fix (as proposed in the original LN whitepaper) but it would have been so much more complex and less efficient that it was well worth fixing it in Bitcoin first, which was the plan anyway even before LN was proposed.