Are two terms where one is without a $\lambda\beta$ normal form unconvertible in $\lambda\beta$?
It is trivially the case that two terms can be $\beta$-convertible even though they do not have a normal form.

Consider:

$$((\lambda x.x)\ (\lambda x.x\ x))\ (\lambda x.x\ x)$$

and

$$(\lambda x.x\ x)\ (\lambda x.x\ x).$$

In one step we have 
$$((\lambda x.x)\ (\lambda x.x\ x))\ (\lambda x.x\ x)\to (\lambda x.x\ x)\ (\lambda x.x\ x)$$