An interesting question, known as the Gaussian moat problem, asks whether it is possible to walk to infinity on Gaussian primes with steps of bounded length. Our work examines a similar situation in the real quadratic integer ring Z[√2] whose primes cluster near the asymptotes y = ± x/√2 as compared to Gaussian primes, which cluster near the origin. We construct a probabilistic model of primes in Z[√2] by applying the prime number theorem and a combinatorial theorem for counting the number of lattice points whose absolute values of their norms are at most r². We then prove that it is impossible to walk to infinity if the walk remains within some bounded distance from the asymptotes. Lastly, we perform a few moat calculations to show that the longest walk is likely to stay close to the asymptotes; hence, we conjecture that there is no walk to infinity on Z[√2] primes with steps of bounded length.