$p$ and $6p+1$ both palindrome - primes. Is $(131/787)$ the only example? $131$ is a palindrome prime as well as $787$ , moreover $6\cdot 131+1=787$.

Are there further examples for a palindrome-prime $p$, such that $6p+1$ is a palindrome-prime as well ?

It is clear that $p$ must have an odd number od digits (since $11$ is the only palindrom-prime with an even number of digits) and the leading digit of $p$ must be $1$ , otherwise $6p+1$ has an even number of digits. 
So, $p$ must have the form $1\cdots 1$ and $6p+1$ must have the form $7\cdots 7$. Upto $10^{10}$ , no further example exits.
 A: This is just a long comment, but it might be of help for anyone wanting to go further.
It makes sense to begin by looking for palindromes $p$, prime or not, but with an odd number of digits (so that it might be prime), such that $6p+1$ is also a palindrome with an odd number of digits (so that it also might be prime). Let's let $p=a_0a_1\ldots a_{n-1}a_na_{n-1}\ldots a_1a_0$ with digits $a_0,a_1,\ldots,a_n\in\{0,1,2,\ldots,9\}$ and $6p+1=b_0b_1\ldots b_{n-1}b_nb_{n-1}\ldots b_1b_0$ with digits $b_0,b_1,\ldots,b_n\in\{0,1,2,\ldots,9\}$.  
As the OP notes, for $6p+1$ to have an odd number of digits, we must have $a_0=1$, in which case $b_0=7$.  But that requires a carry of exactly $1$ from the multiplication $6a_1$, which means $a_1\in\{2,3\}$.  In particular we get the three-digit pairs
$$(121,727)\quad\text{and}\quad(131,787)$$
In general, in order for the carries to work to produce a palindrome, we must have $a_i\in\{0,1\}$ if $i$ is even and $a_i\in\{2,3\}$ if $i$ is odd; moreover, every such choice produces a palindromic pair $(p,6p+1)$. Thus, for example, the five-digit pairs are
$$(12021,72127),\quad(12121,72727),\quad(13031,78187),\quad(13131,78787)$$
Among these numbers, only $72727$ and $78787$ are prime; the $p$ values are all composite. (In particular, $12021$ and $13131$ are easily identified as multiples of $3$.) 
The fact that there are only $2^n$ candidates for a palindromic pair of $2n+1$-digit primes should streamline the search for larger examples considerably. The $11$-digit pair $(12130303121, 72781818727)$ reported by Enzo Creti, for example, is one of only $32$ possible $11$-digit pairs, many of which can be ignored as obvious multiples of $3$.  My guess would be that prime pairs will pop up periodically; perhaps someone can provide a heuristic estimate for their frequency (or an argument to the contrary, that they should eventually cease).
