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In this paper authors for the first time introduce the concept of Neutrosophic Quadruple (NQ) vector spaces and Neutrosophic Quadruple linear algebras and study their properties. Most of the properties of vector spaces are true in case of Neutrosophic Quadruple vector spaces. Two vital observations are, all quadruple vector spaces are of dimension four, be it defined over the field of reals <i>R</i> or the field of complex numbers <i>C</i> or the finite field of characteristic <i>p</i>, <inline-formula> <math display="inline"> <semantics> <msub> <mi>Z</mi> <mi>p</mi> </msub> </semantics> </math> </inline-formula>; <i>p</i> a prime. Secondly all of them are distinct and none of them satisfy the classical property of finite dimensional vector spaces. So this problem is proposed as a conjecture in the final section.