In the past several decades, quantum phase transition and the associated fluctuations have emerged as a major challenge to our understanding of condensed matter. Such transition is tuned by an external parameter such as pressure, chemical doping or magnetic field. The transition point, called quantum critical point (QCP), is only present at absolute zero temperature (T), but its influence (quantum criticality), is spread to nonzero temperature region. Quite often, new stable orders of matter, such as superconductivity, emerge around the QCP, whose relationship with the quantum fluctuations is one of the most important issues. Iron-pnictide superconductors are the second class of high-temperature superconductor family whose phase diagram is very similar to the first class, the copper-oxides. Superconductivity emerges in the vicinity of exotic orders, such as antiferromagnetic, structural or nematic order. Therefore, iron-pnictides provide us a very good opportunity to study quantum criticality. Here we review nuclear magnetic resonance (NMR) study on the coexistence of states and quantum critical phenomena in both hole-doped system Ba1-xKxFe2As2 as well as electron-doped systems BaFe2-xNixAs2 and LaFeAsO1-xFx. Firstly, we found that the 75As NMR spectra split or are broadened for H//c-axis, and shift to a higher frequency for H//ab-plane below a certain temperature in the underdoped region of both hole-doped Ba1-xKxFe2As2 and electron-doped BaFe2-xNixAs2, which indicate that an internal magnetic field develops along the c-axis due to an antiferromagnetic order. Upon further cooling, the spin-lattice relaxation rate 1/T1 measured at the shifted peak shows a distinct decrease below the superconducting critical temperature Tc. These results show unambiguously that the antiferromagnetic order and superconductivity coexist microscopically, which is the essential condition of magnetic QCP. Moreover, the much weaker T-dependence of 1/T1 in the superconducting state compared with the optimal doping sample suggests that the coexisting region is an unusual state and deserves further investigation. Secondly, we conducted transport measurements in electron-doped BaFe2-xNixAs2 system, and found a T-linear resistivity at two critical points. One is at the optimal doping xc1 = 0.10, while the other is in the overdoped region xc2 = 0.14. We found that 1/T1 is nearly T-independent above Tc at xc1 where TN =0, which indicates that xc1 is a magnetic QCP and the observed T-linear resistivity is due to the quantum fluctuation. We find that 1/T1 close to the optimal doping in both Ba1-xKxFe2As2 and LaFeAsO1-xFx also shows a similar behavior as in BaFe2-xNixAs2. The results suggest that superconductivity in these compounds is strongly tied to the quantum antiferromagnetic spin fluctuation. We further studied the structural transition in BaFe2-xNixAs2 by NMR. Since the a-axis and b-axis are not identical below the nematic structural transition temperature Ts, the electric field-gradient becomes asymmetric. Therefore the NMR satellite peaks associated with nuclear spin I=3/2 of 75As split for a twinned single crystal, when the external magnetic field is applied along a- or b-axis. We were able to track the nematic structural transition up to x=0.12. The Ts extrapolates to zero at x=0.14 which suggests that xc2 is a QCP associated with a nematic structural phase transition and the T-linear resistivity at xc2 is therefore due to the QCP. No existing theories can explain such behavior of the resistivity and we call for theoretical investigations in this regard.