This paper is devoted to the study of quantum dissipation in nuclear cluster decay phenomena within the framework of the Lindblad approach to quantum open systems. Typical examples of nuclear decay phenomena, such as cluster radioacivity, spontaneous cold fission of actinide nuclei or quasi-fission of super-heavy nuclei can be described by considering the tunnelling of a metastable state across a piecewise quadratic potential. Two study cases are considered: (i) a harmonic well smoothly joined to an inverted parabola which simulates the barrier and (ii) the inverted parabola of case (i) is continued with a second harmonic well. The width and depth of the second harmonic oscillator well are varied over a wide range of values in order to encompass particular cases of tunnelling in the double-well potential. The evolution of the averages and covariances of the quantum sub-system is studied in both under- and over-damped regimes. For a Gaussian initial wavepacket we compute the tunnelling probability for different values of the friction coefficient and fixed values of the diffusion coefficients. The ansatz used for these coefficients corresponds to the case with temperature T = 0.