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This paper uses the Noether symmetry approach to examine the viable and stable traversable wormhole solutions in the framework of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="script">R</mi><mo>,</mo><msup><mi mathvariant="script">T</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> theory, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula> is the Ricci scalar and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="script">T</mi><mn>2</mn></msup><mo>=</mo><msub><mi mathvariant="script">T</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><msup><mi mathvariant="script">T</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup></mrow></semantics></math></inline-formula> is the self-contraction of the stress–energy tensor. For this purpose, we consider a specific model of this modified theory to obtain the exact solutions of the Noether equations. Further, we formulate the generators of the Noether symmetry and first integrals of motion. We analyze the presence of viable and stable traversable wormhole solutions corresponding to different redshift functions. In order to determine whether this theory provides physically viable and stable wormhole geometry or not, we check the graphical behavior of the null energy constraint, causality condition and adiabatic index for an effective stress–energy tensor. It is found that viable and stable traversable wormhole solutions exist in this modified theory.