Adding to the other answers, the questions leave a few things open to interpretation. Therefore, there are many ways to answer the question. The main thing that isn't clearly stated in the question is whether we think of a keyed hash function. Also, the quote in the question only considers random inputs (assumedly of a given length?). This isn't a common restriction for hash functions, and it's unclear who chooses the input. But we can consider different options.
The common understanding of the property you look after is that the hash function behaves like a random oracle (a random function. But formally characterizing this appears to be tricky. In particular, indistinguishability from a random oracle is not a sound characterization if the hash function is unkeyed. I'll come back to this later.
The hash function is keyed
In this scenario, the property is akin to the undetectability notion mentioned in this answer, which resembles a pseudo-random generator notion but for keyed primitives. Alternatively, since the function is keyed, we may also discuss indistinguishability from a random function. The keyed hash has PRF-like properties even if the inputs aren't random.
The hash function is unkeyed
All the notions mentioned make sense only when the key is hidden from the adversary or when the adversary doesn't choose the inputs and only sees the outputs and must distinguish them from a truly random value. For unkeyed hash functions, if the adversary doesn't select the input, but the challenger chooses random inputs, we are again in a PRG-like scenario, but for shrinking outputs (PRG are deterministic systems).
Indistinguishability from a random function: Sadly, a single (unkeyed) hash function cannot be indistinguishable from a random oracle. The reason is that there are schemes that are secure in the random oracle model but that are always insecure when using a specific hash function. One such example is Canetti, Goldreich and Halevi's "pathological" signature scheme. This means that these schemes essentially become distinguishers.
Indifferentiability: Indistinguishability is not a formalism that is sound for a given, fixed hash function. Indifferentiability has been used to characterize what it is to behave like a random oracle. Today, it is also central for designing and analyzing hash functions (Coron et al.). At a high level, it says that assuming an idealization of the compression function or the permutation, an indifferentiable hash function behaves like a random oracle and can be used in most scenarios where a random oracle is expected without breaking the security. A downside to this is that we need to idealize a building block.