It is a bit tough because both of these reacts with acid so we can not use acid dissolution method. How about TG (Thermogravimetric analsis)?

Well, I just conducted  a brief research. According to this paper

http://www.ejmpep.com/halikia_et.al.pdf

CaCO3 starts to decompose after 635 degree C. Wikipedia says CaOH starts to decompose at 580 degree C (please try to find more reliable reference). So if you allow your mixture to decompose at 600 C for several hours and measure the lost weight, this equals H2O decomposed from Ca(HO)2,  The reaction goes Ca(OH)2 --> CaO + H2O. Again allow the mixture to decompose at 900 C. The lost weight measured is the weight of the lost CO2. The reaction here goes CaCO3--> CaO+ CO2. when you have these values. I think the amount of respective CaOH and CaCO3 can be evaluated from the molar calculations. 

XRF may also be another option. You can quantify the amount of Ca, C and O. But there are many other things to consider before doing this experiment (particularly when you like to quantify C). If you think this would be a better option let me know. I will describe you the method.

To: Kenneth M Towe - Yes sir, I tried with XRD and the pattern confirms that the presence of calcium carbonate and calcium hydroxide but in order to quantify the amount, experts suggested to perform Rietveld refinement analysis. But still I am searching for data with respect to above analysis, as it is very difficult to get data and to run the analysis also.

To: Debendra Neupane - Yes sir, I did TGA and interpretation for the data has to be done. Once it is completed let you know the results. XRF gives results only for metal atoms for eg. total Ca, total Mg, total Al, etc., whatever may be the compounds, if I am wrong please correct me.

Thank you for your valuable suggestions.

Good luck. Yes, XRF only gives the mass % of the elements. 

I think that the simplest way is classical acid-base titration with two indicators. If you take hydrochloric acid with phenolphthalein, you can titrate calcium hydroxide and calcium carbonate up to Ca(HCO3)2, OH -+ H+= H2O CO3 + H+ = HCO3- . Then you add methyl orange and continue the titration of the remaining part of CaCO3 . HCO3- + H== H2CO3 Than you can calculate the quantity of the both components

I agree with Sergei. Titration with the two indicators Is the simplest method if you are concerned with the concentrations of each.

To Sergei and Fredrick,

Dear Sir,

Thank you for your suggestions, Titration with two indicators results will give the amount of hydroxide and carbonates, how will you ensure that it comes from CaCO3 and Ca(OH)2, however that mixture contains other metals like Mg, Na, etc.

First, you did not inform about presence of other metals.

Second, in what kind of chemical compounds (speciation) other metals can present (exist)?

Third, if they exist, so Na can titrated together with Ca(OH)2 and then you can determine it  simply by Flame Atomic Emission Spectrometry. Mg(OH)2 will titrated with Ca(OH)2 together and can be determined photometrically with need organic reagent or by    F-AES. Then you can subtract Mg and Na from whole Me(OH)2 b MeCO3 determined by titration.  If other metals exist as other salt they cannot interact with HCl.

 You can spend protolytic titration by hydrochloric acid with two indicator: methyl orange and phenolphthalein . With phenolphthalein titrated entire hydroxide , and only half of carbonate ions with  methyl orange. Further it is possible to calculate the mountn of carbonate ions and hydroxide

To Karpov Sergey

Dear Sir,

Thank you for your suggestions, Titration with two indicators results will give the amount of hydroxide and carbonates, how will you ensure that it comes from CaCO3 and Ca(OH)2, however that mixture contains other metals like Mg, Na, etc.