Is the convolution of independent normal and gamma also a Pearson distribution? The answer here What is the convolution of a normal distribution with a gamma distribution? gives the pdf of convolution of normal and gamma random variables. Is there a known random variable to which this pdf corresponds to?
 A: 
Is there a known random variable to which this pdf corresponds to?

For any valid probability density function you can define a random variable with that distribution.  Similarly, for any well-defined random variable there is a probability distribution that describes its behaviour.  In the case of the distribution in the linked question, you can form a random variable $W = X+Y$ which is the sum of independent random variables $X \sim \text{N}(\mu, \sigma^2)$ and $Y \sim \text{Ga}(\alpha, \beta)$, and the distribution of $W$ will be the convolution of a normal distribution and a gamma distribution.
I think what you are really asking is whether the form of distribution arising in the linked question is a family of distributions that is commonly used and denoted with a particular name.  I do not recognise it and I have been unable to find any obvious hits in some basic literature searches.  I suspect that this is a family of distributions that has not received much interest in the statistical literature.  There are few contexts where it would make sense to add a normal random variable to a gamma random variable, and since the resulting distribution is extremely messy, I doubt that it constitutes a named and studied family of distributions.  I'm open to being corrected on this pointif anyone else has heard of it.
A: I have been trying to look into this as well. There are three reasons why I find the addition of Normal and Gamma distributions interesting:
(1) Uncertain observations of gamma variables will have a Normal component added to them
(2) It's very easy to come up with a Normal-Gamma convolution that fits the first four moments (mean, variance, skewness, kurtosis) arbitrarily well. One can easily fit a Gamma distribution to the skewness and kurtosis, and as long as the variance of that gamma distribution isn't larger than the overall moment, the Normal distribution just fits the leftover values of mean and variance.
(3) It seems that adding Gamma variables with differing scale parameters will result in the first four moments being fittable to a Normal-Gamma convolution.
While I never found a convenient closed-form solution, it appears that there are quadrature shortcuts for convolving BOTH Normal and Gamma distributions.
Gauss-Laguerre for convolving against a Gamma, and Gauss-Hermite for convolving against a Normal. There's even Julia packages that handle this.
https://juliaapproximation.github.io/FastGaussQuadrature.jl/stable/gaussquadrature/
